Séminaire Lotharingien de Combinatoire, 84B.15 (2020), 12 pp.

Valentin Buciumas, Travis Scrimshaw and Katherine Weber

Colored Five-Vertex Models and Lascoux Polynomials and Atoms

Abstract. We construct an integrable colored five-vertex model whose partition function is a Lascoux atom based on the five-vertex model of Motegi and Sakai and the colored five-vertex model of Brubaker, the first author, Bump, and Gustafsson. We then modify this model in two different ways to construct a Lascoux polynomial, yielding the first known proven combinatorial interpretation of a Lascoux polynomial and atom. Using this, we prove a conjectured combinatorial interpretation in terms of set-valued tableaux of a Lascoux polynomial and atom due to Pechenik and the second author. We also prove the combinatorial interpretation of the Lascoux atom using set-valued skyline tableaux of Monical.


Received: November 20, 2019. Accepted: February 20, 2020. Final version: April 30, 2020.

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